Keith Devlin teaches the theorem that there are infinitely many primes in his course on Mathematical Thinking.

http://www.youtube.com/watch?v=AbZgIj4piv0

Devlin’s proof is easy to follow. He has a good style. Viewers may wish to compare his method to that of the Khan Academy video style with Wacom.

http://www.khanacademy.org/math/arithmetic/factors-multiples

The above are Khan Academy videos relating to simple number theory.

What grade level could we teach the theorem including its proof that there are infinitely many primes?

According to Marco Flores at Zunal.com this is 6th to 8th grade level.

http://zunal.com/export_pdf.php?w=39413

Page 12 of following discusses grade 6 to 8 standards in math.

http://math.arizona.edu/~wmc/Talks/2012_04_22_ICME_slides.pdf

A reference on teaching the infinitely many primes is here.

http://ermeweb.free.fr/CERME5b/WG4.pdf

==

WORKING GROUP 4. Argumentation and proof 536

Argumentation and proof 537

Maria Alessandra Mariotti, Kirsti Hemmi, Viviane Durrand Guerrier

Indirect proof: An interpreting model 541

Samuele Antonini, Maria Alessandra Mariotti

Mathematics learning and the development of general deductive reasoning 551

Michal Ayalon, Ruhama Even

How to decide? Students’ ways of determining the validity of mathematical statements 561

Orly Buchbinder, Orit Zaslavsky

Cabri’s role in the task of proving within the activity of building part of an axiomatic

system 571

Leonor Camargo, Carmen Samper, Patricia Perry

Some remarks on the theorem about the infinity of prime numbers 581

Ercole Castagnola, Roberto Tortora

Proofs problems in elementary number theory: Analysis of trainee teachers’ productions 591

Annalisa Cusi, Nicolina A. Malara

Relationship between beginner teachers in mathematics and the mathematical concept of

implication 601

Virginie Deloustal-Jorrand

Using the Van Hiele theory to analyse the teaching of geometrical proof at grade 8 in

Shanghai 612

Liping Ding, Keith Jones

Analysis of conjectures and proofs produced when learning trigonometry 622

Jorge Fiallo, Angel Gutiérrez

Analysis of the teacher’s arguments used in the didactical management of a problem

solving situation 633

Patrick Gibel

Structural relationships between argumentation and proof in solving open problems in

algebra 643

Bettina Pedemonte

Mathematical proof: Teachers’ beliefs and practices 653

Antonis Sergis

The mental models theory of deductive reasoning: Implications for proof instruction 665

Andreas J. Stylianides, Gabriel J. Stylianides

Reviewing textbook proofs in class: A struggle between proof structure, components and

details 675

Stine Timmermann

=Page 47 Infinitely Many Primes proof

“We want to begin comparing three versions of the proof: the original Euclid’s one

(Proposition 20, book IX of Elements), as reported in (Heath, 1956); the “modern”

(1925) version of the same author; and that used today in mathematics texts.” page 49.

=Page 49 excerpt

THE THEOREM ON THE INFINITY OF PRIME NUMBERS

The theorem on the infinity of prime numbers is one of the most famous and of

the most “beautiful” in the history of mathematics. Several proofs have been

produced (see for instance the website (3)), but the best known, modelled on Euclid’s

original proof, is surely the most easily understood, a striking example of simplicity

and elegance. In spite of that, this proof appears much more obscure for students than

we could think at first sight. A deep and careful analysis of the proof and of its

didactical implications is presented in (Polya, 1973). After that, many authors have

focused their attention on the difficulties involved in the contradiction argument

employed in the proof, e.g. (Reid & Dobbin, 1998). Other authors have underlined

the logical subtleties, all but easy to be understood, involved in such kind of

reasoning (Antonini, 2003), (Antonini & Mariotti, 2006); or the necessity to enter an

“imaginary” world, where the usual rules of logic can be put in doubt (Leron, 1985).

(For more references, see the quoted papers). In particular, Leron notes how the

“distance” between the assumption a contrario and the conclusion causes the total

loss of all the constructions performed in the intermediate steps, erroneously

perceived as meaningless.

=

==

They also consider this problem:

==Page 59 of pdf

PRESENTING THE PROPOSED PROBLEM

The problem at stake is the following: “Suppose that a is a non null natural number.

If a is divisible neither by 2 nor by 3, then a*a-1 is divisible by 24”. This problem is

taken from the textbook, aimed at 15-16 years old students, “Matematica come

scoperta” (“Mathematics as discovery”) by G. Prodi (1979). This textbook was

thought and written in a research-based environment and it is still very innovative.

…

Page 60

METHODOLOGY

The problem was given to 54 trainees with different university backgrounds (27

mathematics graduates, 3 physics graduates and the remaining 24 biology, geology,

chemistry and natural sciences graduates) in the initial phase of the Mathematics

Education training courses. Trainees were supposed to solve, in 45 minutes, the

assigned problem, describing the different proving strategies they tried to follow in

the solution process and pointing out both obstacles and difficulties they met.

==

Results follow

==

Let’s Play Math gets into it.

http://letsplaymath.net/2010/03/19/math-teachers-at-play-24/

It links here:

http://www.johndcook.com/blog/2010/02/13/euclids-proof-that-there-are-infinitely-many-primes/

They try to fit the proof to the Twitter span of 140 characters.

Kentucky Grade 6 standards

http://www.ixl.com/math/standards/kentucky/grade-6

“infinitely many primes” “grade level”

You can add Euclid to this search.

=

Proof by Contradiction video

Infinitely many primes 8 minutes in.

http://www.youtube.com/watch?v=9bRsR8z7r8M

He goes through the steps more slowly than Devlin.

=

http://www.youtube.com/watch?v=Jq2031V-SLU&feature=related

The above does theorem (n+1)^-n^2 = n+(n+1)

= 2n+1 = 2n’-1

Notice how easier it is if we use prime notation.

n’ n’ = (n+1)n’ = nn’ + n’ = n (n+1) + n’ = nn + n + n’

so

n’ n’ – nn = n + n’